The core of "partial pressure" is that gases can exist in a mixture (like air, being 21% oxygen and 79% nitrogen) but the gases behave as though they don't know the others are there. So, when you put gases into contact with a liquid, the gases dissolve into the liquid according to the pressure of each individual gas. If you have water sitting in contact with atmospheric air, the amount of oxygen that will dissolve into the liquid is the same amount that would dissolve if you put the water in a container and put only .21 ATA of pure O2 in it.
When you descend, the gas you are breathing is more dense, so any given volume contains a great many more molecules of each gas. That's like putting a gas at a higher pressure in contact with the liquid. So, at 33 feet, you have 21% oxygen, but the total pressure isn't 1 ATA any more, it's two. So the total pressure of oxygen is .42 ATA, and the amount of oxygen that will dissolve into your blood is the amount that would happen if you put pure oxygen at .42 ATA in contact with that blood. Each gas sorts itself the same way; they don't affect one another, so you can consider them as totally independent.
Because of this, you can use a shorthand for describing the concentration of dissolved oxygen (or any gas) in a liquid. You don't have to measure millimoles per liter, or nanograms per deciliter. You can just use the pressure that would generate that concentration, and say, ".42 ATA" instead, and everyone will know what that means.
This gets very important in two ways. One is that oxygen becomes toxic at high concentrations (remember, we will describe concentrations as partial pressures). Above 1.4 ATA, you start to have a risk of seizures. The absolute risk is not well-defined, because it varies between people and in any given person from day to day, but the danger of seizing underwater is so great that we simply arbitrarily set a limit beyond which it is unwise to go, at least for very long.
The other way is that nitrogen at high concentrations is an anesthetic, manifested in recreational depths by nitrogen narcosis. You have to have a moderate concentration of nitrogen to notice the effects -- but again, we describe concentrations of gases as "partial pressures", meaning the fraction of the total pressure represented by THAT gas. A lot of people like to keep their nitrogen concentrations at, or lower than that of air at 100 feet, which is about 3.2 ATA.
Did that help?